Why don't I get numeric values from this Solve?

Question

Solve[ a^2 + b^2 == c^2 && a < 100 && b < 100 && c < 100, {a, b, c}, Integers]

gives me this result

{{c -> ConditionalExpression[-Sqrt[ a^2 + b^2],
         ((a | b | c) ∈ Integers &&  Sqrt[10000 - a^2] - b > 0  && -99 <= a <= -1) ||
         ((a | b | c) ∈ Integers &&  Sqrt[10000 - a^2] - b > 0  &&   1 <= a <= 99) || 
         ((a | b | c) ∈ Integers &&  Sqrt[10000 - a^2] + b <= 0 && -99 <= a <= -1) ||
         ((a | b | c) ∈ Integers &&  Sqrt[10000 - a^2] + b <= 0 &&   1 <= a <= 99) ||
              ...           ] },   ...  }

where I would expect something like

{{x -> 3, y -> 4, z -> 5}, {x -> 4, y -> 3, z -> 5}, {x -> 5, y -> 12, z -> 13},...}

What am I doing wrong?

Answer 1

This works (it's obvious the same will hold for negative integers):

Reduce[a^2 + b^2 == c^2 && 0 < a < 100 && 0 < b < 100 && 0 < c < 100, {a, b, c}, Integers]

And if you're up for an interesting visualization, here is my take:

ListPlot3D[{a, b, c} /. Solve[a^2 + b^2 == c^2 && 0 < a < 100 && 0 < b < 100 && 
    0 < c < 100, {a, b, c}, Integers], InterpolationOrder -> 0, 
 Mesh -> None, ColorFunction -> "Rainbow", Filling -> Bottom, BoxRatios -> 1]

Answer 2

Bound the variables from both sides if you want to get explicit results

s = Solve[a^2+b^2==c^2&& -100 < a < 100&& -100 < b < 100&& -100 < c < 100, {a,b,c}, Integers]
(* -> {{a -> -99, b -> 0, c -> -99}, {a -> -98, b -> 0, c -> 98} ... *)
ListPointPlot3D[{a, b, c} /. s]

Or you could reduce the domain to positive integers, taken advantage of the problem being quadratic on all vars:

s = Solve[a^2 + b^2 - c^2 == 0 && 
      0 <= a < 100 && 0 <= b < 100 &&  0 <= c < 100, {a, b, c}, Integers]

ListPointPlot3D[{a, b, c} /. s]

Edit

And you can verify that

2 Sum[SquaresR[2, i^2], {i, 0, 99}] -1 == (2 Length@# -Count[#, 0]) &@Flatten[{a, b, c} /. s]
(* True *)

This is a useful reading when solving Diophantine equations

Answer 3

The result we get is correct and one shouldn't expect it in a different form since there are infinitely many solutions and there is no way to enumerate them all. That's why Mathematica returns the result in a symbolic way. This issue appears a bit more obviously when one looks for all solutions in special subclasses, e.g. solutions of this form :

a == 3k, b == 4k, c == 5k for k < 20 && k ∈ Integers, e.g. k == -10^100000. We have to use Reduce instead of Solve in this case :

Reduce[(3 k)^2 + (4 k)^2 == (5 k)^2 && k < 20, k, Integers]

k ∈ Integers && k <= 19

There are infinitely many solutions, however if we add a lower bound for k we'll get a finite numbers of explicit solutions :

Reduce[(3 k)^2 + (4 k)^2 == (5 k)^2 && 15 < k < 20, k, Integers]

k == 16 || k == 17 || k == 18 || k == 19

Let's visualize an interesting part of the solution space to the general problem :

pts = Solve[{ a^2 + b^2 == c^2,
             -300 < a < 100, -300 < b < 100, -300 < c < 100}, {a, b, c}, Integers];
Length[pts]

2345

Show[ ContourPlot3D[ x^2 + y^2 == z^2, {x, -300, 100}, {y, -300, 100}, {z, -300, 100}, 
                    ContourStyle -> Directive[Orange, Opacity[0.3], Specularity[White, 20]], 
                    MeshFunctions -> {#3 &}, Mesh -> 7, PlotPoints -> 100], 
      ListPointPlot3D[ {a, b, c} /. pts, PlotStyle -> Directive[Darker@Red, PointSize[0.004]]]]

These cones (the upper and a part of the lower) denote all real solutions whereas the red points denote all integer sloutions of the sytem : { a^2 + b^2 == c^2, -300 < a < 100, -300 < b < 100, -300 < c < 100}

One might be also interested in integer solutions modulo 100 to find all solutions of this type : a^2 + b^2 == c^2 + 100 k for k ∈ Integers (i.e. solutions satisfy 0 <= a < 100, 0 <= b < 100, 0 <= c < 100 ), there are many more such solutions ( i.e. 17400) :

Solve[ a^2 + b^2 == c^2, {a, b, c}, Modulus -> 100] // Short[#, 2] &

{{a -> 0, b -> 0, c -> 0},  {a -> 0, b -> 0, c -> 10}, {a -> 0, b -> 0, c -> 20},
 {a -> 0, b -> 0, c -> 30},   <<17393>>,     {a -> 99, b -> 90, c -> 49},
 {a -> 99, b -> 90, c -> 51}, {a -> 99, b -> 90, c -> 99}}